发表机构
HUN-REN Alfréd Rényi Institute of Mathematics; Budapest University of Technology and Economics（HUN-REN 阿尔弗雷德·雷尼数学研究所; 布达佩斯技术与经济大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究量子瓦瑟斯坦距离，证明正半定成本算子由纯态距离限制确定，在特定条件下瓦瑟斯坦等距幺半群由维格纳对称性组成，构造相关互逆映射，得出非零各向同性成本与自伴生成元数量关系及相关结论。
AI 中文摘要
我们研究了由德帕尔马和特雷维桑引入的与自伴可观测量族生成的二次成本算子相关的量子瓦瑟斯坦距离。首先表明任意正半定成本算子完全由相应瓦瑟斯坦距离对纯态对的限制确定。在\(d\)维希尔伯特空间上由至多\(d^2 - 1\)个可观测量生成的非零二次成本类中，证明当且仅当该距离在纯态上的酉共轭下不变时，瓦瑟斯坦等距幺半群恰好由维格纳对称性组成。还构造了可观测量生成的二次成本算子与其无迹部分形成的希尔伯特 - 施密特框架型算子之间的显式互逆映射。结果表明成本的各向同性等同于相关希尔伯特 - 施密特算子的紧框架性质。因此，非零各向同性成本至少需要\(d^2 - 1\)个自伴生成元，当且仅当它们的无迹部分构成一个希尔伯特 - 施密特正交基时等号成立。所以几何、表示理论、算子理论和框架理论的对称概念都确定了同一个单参数量子瓦瑟斯坦距离族。
英文摘要
We study the quantum Wasserstein distances of De Palma and Trevisan associated with quadratic costs generated by self-adjoint observables in arbitrary finite dimension. We first show that any positive semidefinite cost operator is determined by the corresponding distance on pairs of pure states. For nonzero observable-generated quadratic costs, we then prove that the full isometry monoid consists exactly of unitary and antiunitary conjugations if and only if the distance is invariant under unitary conjugations on pure states. These conditions are equivalent to isotropy of the cost on the traceless subspace. They are also equivalent to a geometric characterization: two states attain the Wasserstein diameter exactly when their supports are orthogonal. It already suffices to impose this condition on pairs of pure states. We construct explicit mutually inverse maps between observable-generated quadratic costs and Hilbert--Schmidt frame-type operators formed from the traceless parts of their generators. This correspondence also expresses the transport variational problem directly in terms of the frame operator. For every such cost $C$ on a $d$-dimensional Hilbert space, the minimal number of self-adjoint generators equals the rank of its recovered frame operator $Θ(C)$, and is therefore at most $d^2-1$. Nonzero isotropic costs require exactly $d^2-1$ generators, attaining the largest minimal generator number that any observable-generated quadratic cost can require. Their arbitrary generating families give tight frames, while their minimal realizations give scaled Hilbert--Schmidt orthonormal bases of the traceless subspace. These symmetry, frame, and diametral characterizations determine the same one-parameter family of quantum Wasserstein distances.
Comments25 pages. Revised and expanded version. Added a characterization of diameter-attaining pairs for the isotropic Wasserstein distance, clarified the generator-count statements and several proofs, corrected minor errors, and improved the presentation throughout
